Pharmaceutical Process Scale-Up

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PharmaceuticalProcess Scale-Up

edited byMichael Levin

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DRUGS AND THE PHARMACEUTICAL SCIENCES

Executive EditorJames SwarbrickPharmaceuTech, Inc.

Pinehurst, North Carolina

Advisory Board

Larry L. AugsburgerUniversity of Maryland

Baltimore, Maryland

David E. NicholsPurdue UniversityWest Lafayette, Indiana

Douwe D. BreimerGorlaeus Laboratories

Leiden, The Netherlands

Stephen G. SchulmanUniversity of FloridaGainesville, Florida

Trevor M. JonesThe Association of the

British Pharmaceutical IndustryLondon, United Kingdom

Jerome P. SkellyAlexandria, Virginia

Hans E. JungingerLeiden/Amsterdam Center

for Drug ResearchLeiden, The Netherlands

Felix TheeuwesAlza CorporationPalo Alto, California

Vincent H. L. LeeUniversity of Southern California

Los Angeles, California

Geoffrey T. TuckerUniversity of SheffieldRoyal Hallamshire HospitalSheffield, United Kingdom

Peter G. WellingInstitut de Recherche Jouveinal

Fresnes, France

DRUGS AND THE PHARMACEUTICAL SCIENCES

A Series of Textbooks and Monographs

1. Pharmacokinetics, Milo Gibaldi and Donald Perrier2. Good Manufacturing Practices for Pharmaceuticals: A Plan for Total

Quality Control, Sidney H. Willig, Murray M. Tuckerman, and WilliamS. Hitchings IV

3. Microencapsulation, edited by J. R. Nixon4. Drug Metabolism: Chemical and Biochemical Aspects, Bernard Testa

and Peter Jenner5. New Drugs: Discovery and Development, edited by Alan A. Rubin6. Sustained and Controlled Release Drug Delivery Systems, edited by

Joseph R. Robinson7. Modern Pharmaceutics, edited by Gilbert S. Banker and Christopher

T. Rhodes8. Prescription Drugs in Short Supply: Case Histories, Michael A.

Schwartz9. Activated Charcoal: Antidotal and Other Medical Uses, David O.

Cooney10. Concepts in Drug Metabolism (in two parts), edited by Peter Jenner

and Bernard Testa11. Pharmaceutical Analysis: Modern Methods (in two parts), edited by

James W. Munson12. Techniques of Solubilization of Drugs, edited by Samuel H. Yalkowsky13. Orphan Drugs, edited by Fred E. Karch14. Novel Drug Delivery Systems: Fundamentals, Developmental Con-

cepts, Biomedical Assessments, Yie W. Chien15. Pharmacokinetics: Second Edition, Revised and Expanded, Milo

Gibaldi and Donald Perrier16. Good Manufacturing Practices for Pharmaceuticals: A Plan for Total

Quality Control, Second Edition, Revised and Expanded, Sidney H.Willig, Murray M. Tuckerman, and William S. Hitchings IV

17. Formulation of Veterinary Dosage Forms, edited by Jack Blodinger18. Dermatological Formulations: Percutaneous Absorption, Brian W.

Barry19. The Clinical Research Process in the Pharmaceutical Industry, edited

by Gary M. Matoren20. Microencapsulation and Related Drug Processes, Patrick B. Deasy21. Drugs and Nutrients: The Interactive Effects, edited by Daphne A.

Roe and T. Colin Campbell22. Biotechnology of Industrial Antibiotics, Erick J. Vandamme23. Pharmaceutical Process Validation, edited by Bernard T. Loftus and

Robert A. Nash

24. Anticancer and Interferon Agents: Synthesis and Properties, edited byRaphael M. Ottenbrite and George B. Butler

25. Pharmaceutical Statistics: Practical and Clinical Applications, SanfordBolton

26. Drug Dynamics for Analytical, Clinical, and Biological Chemists,Benjamin J. Gudzinowicz, Burrows T. Younkin, Jr., and Michael J.Gudzinowicz

27. Modern Analysis of Antibiotics, edited by Adjoran Aszalos28. Solubility and Related Properties, Kenneth C. James29. Controlled Drug Delivery: Fundamentals and Applications, Second

Edition, Revised and Expanded, edited by Joseph R. Robinson andVincent H. Lee

30. New Drug Approval Process: Clinical and Regulatory Management,edited by Richard A. Guarino

31. Transdermal Controlled Systemic Medications, edited by Yie W.Chien

32. Drug Delivery Devices: Fundamentals and Applications, edited byPraveen Tyle

33. Pharmacokinetics: Regulatory Industrial Academic Perspectives,edited by Peter G. Welling and Francis L. S. Tse

34. Clinical Drug Trials and Tribulations, edited by Allen E. Cato35. Transdermal Drug Delivery: Developmental Issues and Research Ini-

tiatives, edited by Jonathan Hadgraft and Richard H. Guy36. Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms,

edited by James W. McGinity37. Pharmaceutical Pelletization Technology, edited by Isaac Ghebre-

Sellassie38. Good Laboratory Practice Regulations, edited by Allen F. Hirsch39. Nasal Systemic Drug Delivery, Yie W. Chien, Kenneth S. E. Su, and

Shyi-Feu Chang40. Modern Pharmaceutics: Second Edition, Revised and Expanded,

edited by Gilbert S. Banker and Christopher T. Rhodes41. Specialized Drug Delivery Systems: Manufacturing and Production

Technology, edited by Praveen Tyle42. Topical Drug Delivery Formulations, edited by David W. Osborne and

Anton H. Amann43. Drug Stability: Principles and Practices, Jens T. Carstensen44. Pharmaceutical Statistics: Practical and Clinical Applications, Second

Edition, Revised and Expanded, Sanford Bolton45. Biodegradable Polymers as Drug Delivery Systems, edited by Mark

Chasin and Robert Langer46. Preclinical Drug Disposition: A Laboratory Handbook, Francis L. S.

Tse and James J. Jaffe47. HPLC in the Pharmaceutical Industry, edited by Godwin W. Fong and

Stanley K. Lam48. Pharmaceutical Bioequivalence, edited by Peter G. Welling, Francis L.

S. Tse, and Shrikant V. Dinghe49. Pharmaceutical Dissolution Testing, Umesh V. Banakar

50. Novel Drug Delivery Systems: Second Edition, Revised andExpanded, Yie W. Chien

51. Managing the Clinical Drug Development Process, David M. Coc-chetto and Ronald V. Nardi

52. Good Manufacturing Practices for Pharmaceuticals: A Plan for TotalQuality Control, Third Edition, edited by Sidney H. Willig and JamesR. Stoker

53. Prodrugs: Topical and Ocular Drug Delivery, edited by Kenneth B.Sloan

54. Pharmaceutical Inhalation Aerosol Technology, edited by Anthony J.Hickey

55. Radiopharmaceuticals: Chemistry and Pharmacology, edited byAdrian D. Nunn

56. New Drug Approval Process: Second Edition, Revised and Expanded,edited by Richard A. Guarino

57. Pharmaceutical Process Validation: Second Edition, Revised and Ex-panded, edited by Ira R. Berry and Robert A. Nash

58. Ophthalmic Drug Delivery Systems, edited by Ashim K. Mitra59. Pharmaceutical Skin Penetration Enhancement, edited by Kenneth A.

Walters and Jonathan Hadgraft60. Colonic Drug Absorption and Metabolism, edited by Peter R. Bieck61. Pharmaceutical Particulate Carriers: Therapeutic Applications, edited

by Alain Rolland62. Drug Permeation Enhancement: Theory and Applications, edited by

Dean S. Hsieh63. Glycopeptide Antibiotics, edited by Ramakrishnan Nagarajan64. Achieving Sterility in Medical and Pharmaceutical Products, Nigel A.

Halls65. Multiparticulate Oral Drug Delivery, edited by Isaac Ghebre-Sellassie66. Colloidal Drug Delivery Systems, edited by Jrg Kreuter67. Pharmacokinetics: Regulatory Industrial Academic Perspectives,

Second Edition, edited by Peter G. Welling and Francis L. S. Tse68. Drug Stability: Principles and Practices, Second Edition, Revised and

Expanded, Jens T. Carstensen69. Good Laboratory Practice Regulations: Second Edition, Revised and

Expanded, edited by Sandy Weinberg70. Physical Characterization of Pharmaceutical Solids, edited by Harry

G. Brittain71. Pharmaceutical Powder Compaction Technology, edited by Gran Al-

derborn and Christer Nystrm72. Modern Pharmaceutics: Third Edition, Revised and Expanded, edited

by Gilbert S. Banker and Christopher T. Rhodes73. Microencapsulation: Methods and Industrial Applications, edited by

Simon Benita74. Oral Mucosal Drug Delivery, edited by Michael J. Rathbone75. Clinical Research in Pharmaceutical Development, edited by Barry

Bleidt and Michael Montagne

76. The Drug Development Process: Increasing Efficiency and Cost Ef-fectiveness, edited by Peter G. Welling, Louis Lasagna, and UmeshV. Banakar

77. Microparticulate Systems for the Delivery of Proteins and Vaccines,edited by Smadar Cohen and Howard Bernstein

78. Good Manufacturing Practices for Pharmaceuticals: A Plan for TotalQuality Control, Fourth Edition, Revised and Expanded, Sidney H.Willig and James R. Stoker

79. Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms:Second Edition, Revised and Expanded, edited by James W.McGinity

80. Pharmaceutical Statistics: Practical and Clinical Applications, ThirdEdition, Sanford Bolton

81. Handbook of Pharmaceutical Granulation Technology, edited by DilipM. Parikh

82. Biotechnology of Antibiotics: Second Edition, Revised and Expanded,edited by William R. Strohl

83. Mechanisms of Transdermal Drug Delivery, edited by Russell O. Pottsand Richard H. Guy

84. Pharmaceutical Enzymes, edited by Albert Lauwers and SimonScharp

85. Development of Biopharmaceutical Parenteral Dosage Forms, editedby John A. Bontempo

86. Pharmaceutical Project Management, edited by Tony Kennedy87. Drug Products for Clinical Trials: An International Guide to Formula-

tion Production Quality Control, edited by Donald C. Monkhouseand Christopher T. Rhodes

88. Development and Formulation of Veterinary Dosage Forms: SecondEdition, Revised and Expanded, edited by Gregory E. Hardee and J.Desmond Baggot

89. Receptor-Based Drug Design, edited by Paul Leff90. Automation and Validation of Information in Pharmaceutical Pro-

cessing, edited by Joseph F. deSpautz91. Dermal Absorption and Toxicity Assessment, edited by Michael S.

Roberts and Kenneth A. Walters92. Pharmaceutical Experimental Design, Gareth A. Lewis, Didier

Mathieu, and Roger Phan-Tan-Luu93. Preparing for FDA Pre-Approval Inspections, edited by Martin D.

Hynes III94. Pharmaceutical Excipients: Characterization by IR, Raman, and NMR

Spectroscopy, David E. Bugay and W. Paul Findlay95. Polymorphism in Pharmaceutical Solids, edited by Harry G. Brittain96. Freeze-Drying/Lyophilization of Pharmaceutical and Biological Prod-

ucts, edited by Louis Rey and Joan C. May97. Percutaneous Absorption: DrugsCosmeticsMechanismsMetho-

dology, Third Edition, Revised and Expanded, edited by Robert L.Bronaugh and Howard I. Maibach

98. Bioadhesive Drug Delivery Systems: Fundamentals, Novel Ap-proaches, and Development, edited by Edith Mathiowitz, Donald E.Chickering III, and Claus-Michael Lehr

99. Protein Formulation and Delivery, edited by Eugene J. McNally100. New Drug Approval Process: Third Edition, The Global Challenge,

edited by Richard A. Guarino101. Peptide and Protein Drug Analysis, edited by Ronald E. Reid102. Transport Processes in Pharmaceutical Systems, edited by Gordon L.

Amidon, Ping I. Lee, and Elizabeth M. Topp103. Excipient Toxicity and Safety, edited by Myra L. Weiner and Lois A.

Kotkoskie104. The Clinical Audit in Pharmaceutical Development, edited by Michael

R. Hamrell105. Pharmaceutical Emulsions and Suspensions, edited by Francoise

Nielloud and Gilberte Marti-Mestres106. Oral Drug Absorption: Prediction and Assessment, edited by Jennifer

B. Dressman and Hans Lennerns107. Drug Stability: Principles and Practices, Third Edition, Revised and

Expanded, edited by Jens T. Carstensen and C. T. Rhodes108. Containment in the Pharmaceutical Industry, edited by James P.

Wood109. Good Manufacturing Practices for Pharmaceuticals: A Plan for Total

Quality Control from Manufacturer to Consumer, Fifth Edition, Revisedand Expanded, Sidney H. Willig

110. Advanced Pharmaceutical Solids, Jens T. Carstensen111. Endotoxins: Pyrogens, LAL Testing, and Depyrogenation, Second

Edition, Revised and Expanded, Kevin L. Williams112. Pharmaceutical Process Engineering, Anthony J. Hickey and David

Ganderton113. Pharmacogenomics, edited by Werner Kalow, Urs A. Meyer, and Ra-

chel F. Tyndale114. Handbook of Drug Screening, edited by Ramakrishna Seethala and

Prabhavathi B. Fernandes115. Drug Targeting Technology: Physical Chemical Biological Methods,

edited by Hans Schreier116. DrugDrug Interactions, edited by A. David Rodrigues117. Handbook of Pharmaceutical Analysis, edited by Lena Ohannesian

and Anthony J. Streeter118. Pharmaceutical Process Scale-Up, edited by Michael Levin119. Dermatological and Transdermal Formulations, edited by Kenneth A.

Walters120. Clinical Drug Trials and Tribulations: Second Edition, Revised and

Expanded, edited by Allen Cato, Lynda Sutton, and Allen Cato III121. Modern Pharmaceutics: Fourth Edition, Revised and Expanded, edi-

ted by Gilbert S. Banker and Christopher T. Rhodes122. Surfactants and Polymers in Drug Delivery, Martin Malmsten123. Transdermal Drug Delivery: Second Edition, Revised and Expanded,

edited by Richard H. Guy and Jonathan Hadgraft

124. Good Laboratory Practice Regulations: Second Edition, Revised andExpanded, edited by Sandy Weinberg

125. Parenteral Quality Control: Sterility, Pyrogen, Particulate, and Pack-age Integrity Testing: Third Edition, Revised and Expanded, MichaelJ. Akers, Daniel S. Larrimore, and Dana Morton Guazzo

126. Modified-Release Drug Delivery Technology, edited by Michael J.Rathbone, Jonathan Hadgraft, and Michael S. Roberts

127. Simulation for Designing Clinical Trials: A Pharmacokinetic-Pharma-codynamic Modeling Perspective, edited by Hui C. Kimko and Ste-phen B. Duffull

128. Affinity Capillary Electrophoresis in Pharmaceutics and Biopharma-ceutics, edited by Reinhard H. H. Neubert and Hans-Hermann Rt-tinger

129. Pharmaceutical Process Validation: An International Third Edition,Revised and Expanded, edited by Robert A. Nash and Alfred H.Wachter

130. Ophthalmic Drug Delivery Systems: Second Edition, Revised andExpanded, edited by Ashim K. Mitra

131. Pharmaceutical Gene Delivery Systems, edited by Alain Rolland andSean M. Sullivan

ADDITIONAL VOLUMES IN PREPARATION

Biomarkers in Clinical Drug Development, edited by John Bloom

Pharmaceutical Inhalation Aerosol Technology: Second Edition, Re-vised and Expanded, edited by Anthony J. Hickey

Pharmaceutical Extrusion Technology, edited by Isaac Ghebre-Sellas-sie and Charles Martin

Pharmaceutical Compliance, edited by Carmen Medina

To my wife Sonia,my children Hanna, Daniela, Ilan, and Emanuel,

and to the memory of my parents.

vPreface

Pharmaceutical Process Scale-Up deals with a subject both fascinating and vi-tally important for the pharmaceutical industrythe procedures of transferringthe results of R&D obtained on laboratory scale to the pilot plant and finally toproduction scale. The primary objective of the text is to provide insight into thepractical aspects of process scale-up. As a source of information on batch en-largement techniques, it will be of practical interest to formulators, process engi-neers, validation specialists and quality assurance personnel, as well as productionmanagers. The book also provides interesting reading for those involved in tech-nology transfer and product globalization.

Since engineering support and maintenance are crucial for proper scale-upof any process, Chapter 10 discusses plant design and machinery maintenance is-sues. Regulatory aspects of scale-up and postapproval changes are addressedthroughout the book but more specifically in Chapter 11. A diligent attempt wasmade to keep all references to FDA regulations as complete and current as possi-ble. Although some theory and history of process scale-up are discussed, knowl-edge of physics or engineering is not required of the reader since all theoreticalconsiderations are fully explained.

Michael Levin

Introduction

Scale-up is generally defined as the process of increasing the batch size. Scale-upof a process can also be viewed as a procedure for applying the same process todifferent output volumes. There is a subtle difference between these two defini-tions: batch size enlargement does not always translate into a size increase of theprocessing volume.

In mixing applications, scale-up is indeed concerned with increasing thelinear dimensions from the laboratory to the plant size. On the other hand, pro-cesses exist (e.g., tableting) for which scale-up simply means enlarging theoutput by increasing the speed. To complete the picture, one should point outspecial procedures (especially in biotechnology) in which an increase of thescale is counterproductive and scale-down is required to improve the qualityof the product.

In moving from R&D to production scale, it is sometimes essential to havean intermediate batch scale. This is achieved at the so-called pilot scale, which isdefined as the manufacturing of drug product by a procedure fully representativeof and simulating that used for full manufacturing scale. This scale also makespossible the production of enough product for clinical testing and samples formarketing. However, inserting an intermediate step between R&D and productionscales does not in itself guarantee a smooth transition. A well-defined process maygenerate a perfect product in both the laboratory and the pilot plant and then failquality assurance tests in production.

Imagine that you have successfully scaled up a mixing or a granulating pro-cess from a 10-liter batch to a 75-liter and then to a 300-liter batch. What exactlyhappened? You may say, I got lucky. Apart from luck, there had to be some

vii

physical similarity in the processing of the batches. Once you understand whatmakes these processes similar, you can eliminate many scale-up problems.

A rational approach to scale-up has been used in physical sciences, viz. fluiddynamics and chemical engineering, for quite some time. This approach is basedon process similarities between different scales and employs dimensional analy-sis that was developed a century ago and has since gained wide recognition inmany industries, especially in chemical engineering [1].

Dimensional analysis is a method for producing dimensionless numbers thatcompletely characterize the process. The analysis can be applied even when theequations governing the process are not known. According to the theory of mod-els, two processes may be considered completely similar if they take place in sim-ilar geometrical space and if all the dimensionless numbers necessary to describethe process have the same numerical value [2]. The scale-up procedure, then, issimple: express the process using a complete set of dimensionless numbers, andtry to match them at different scales. This dimensionless space in which the mea-surements are presented or measured will make the process scale invariant.

Dimensionless numbers, such as Reynolds and Froude numbers, are frequentlyused to describe mixing processes. Chemical engineers are routinely concerned withproblems of water-air or fluid mixing in vessels equipped with turbine stirrers inwhich scale-up factors can be up to 1:70 [3]. This approach has been applied to phar-maceutical granulation since the early work of Hans Leuenberger in 1982 [4].

One way to eliminate potential scale-up problems is to develop formulationsthat are very robust with respect to processing conditions. A comprehensivedatabase of excipients detailing their material properties may be indispensable forthis purpose. However, in practical terms, this cannot be achieved without somemeans of testing in a production environment, and, since the initial drug substanceis usually available only in small quantities, some form of simulation is requiredon a small scale.

In tableting applications, the process scale-up involves different speeds ofproduction in what is essentially the same unit volume (die cavity in which thecompaction takes place). Thus, one of the conditions of the theory of models (sim-ilar geometric space) is met. However, there are still kinematic and dynamic pa-rameters that need to be investigated and matched for any process transfer. One ofthe main practical questions facing tablet formulators during development andscale-up is whether a particular formulation will sustain the required high rate ofcompression force application in a production press without lamination or cap-ping. Usually, such questions are never answered with sufficient credibility, es-pecially when only a small amount of material is available and any trial-and-errorapproach may result in costly mistakes along the scale-up path.

As tablet formulations are moved from small-scale research presses to high-speed machines, potential scale-up problems can be eliminated by simulation ofproduction conditions in the formulation development lab. In any process transfer

viii Introduction

from one tablet press to another, one may aim to preserve mechanical propertiesof a tablet (density and, by extension, energy used to obtain it) as well as itsbioavailability (e.g., dissolution that may be affected by porosity). A scientificallysound approach would be to use the results of the dimensional analysis to modela particular production environment. Studies done on a class of equipment gener-ally known as compaction simulators or tablet press replicators can be designed tofacilitate the scale-up of tableting process by matching several major factors, suchas compression force and rate of its application (punch velocity and displace-ment), in their dimensionless equivalent form.

Any significant change in a process of making a pharmaceutical dosage formis a regulatory concern. Scale-Up and Postapproval Changes (SUPAC) are of specialinterest to the FDA, as is evidenced by a growing number of regulatory documentsreleased in the past several years by the Center for Drug Evaluation and Research(CDER), including Immediate Release Solid Oral Dosage Forms (SUPAC-IR),Modified Release Solid Oral Dosage Forms (SUPAC-MR), and Semisolid DosageForms (SUPAC-SS). Additional SUPAC guidance documents being developed in-clude: Transdermal Delivery Systems (SUPAC-TDS), Bulk Actives (BACPAC),and Sterile Aqueous Solutions (PAC-SAS). Collaboration between the FDA, thepharmaceutical industry, and academia in this and other areas has recently beenlaunched under the framework of the Product Quality Research Institute (PQRI).

Scale-up problems may require postapproval changes that affect formulationcomposition, site, and manufacturing process or equipment (from the regulatorystandpoint, scale-up and scale-down are treated with the same degree of scrutiny).In a typical drug development cycle, once a set of clinical studies has been com-pleted or an NDA/ANDA has been approved, it becomes very difficult to changethe product or the process to accommodate specific production needs. Such needsmay include changes in batch size and manufacturing equipment or process.

Postapproval changes in the size of a batch from the pilot scale to larger orsmaller production scales call for submission of additional information in the ap-plication, with a specific requirement that the new batches are to be produced us-ing similar test equipment and in full compliance with CGMPs and the existingSOPs. Manufacturing changes may require new stability, dissolution, and in vivobioequivalence testing. This is especially true for Level 2 equipment changes(change in equipment to a different design and different operating principles) andthe process changes of Level 2 (process changes, e.g., in mixing times and oper-ating speeds within application/validation ranges) and Level 3 (change in the typeof process used in the manufacture of the product, such as from wet granulation todirect compression of dry powder).

Any such testing and accompanying documentation are subject to FDA ap-proval and can be very costly. In 1977, the FDAs Office of Planning and Evalu-ation (OPE) studied the impact on industry of the SUPAC guidance, including itseffects on cost. The findings indicated that the guidance resulted in substantial

Introduction ix

savings because it permitted, among other things, shorter waiting times for sitetransfers and more rapid implementation of process and equipment changes, aswell as increases in batch size and reduction of quality control costs.

In early development stages of a new drug substance, relatively little infor-mation is available regarding its polymorphic forms, solubility, and other aspects.As the final formulation is developed, changes to the manufacturing process maychange the purity profile or physical characteristics of the drug substance and thuscause batch failures and other problems with the finished dosage form.

FDA inspectors are instructed to look for any differences between the pro-cess filed in the application and the process used to manufacture the bio/clinicalbatch. Furthermore, one of the main requirements of a manufacturing process isthat it will yield a product that is equivalent to the substance on which the bio-study or pivotal clinical study was conducted. Validation of the process develop-ment and scale-up should include sufficient documentation so that a link betweenthe bio/clinical batches and the commercial process can be established. If the pro-cess is different after scale-up, the company has to demonstrate that the productproduced by a modified process will be equivalent, using data such as granulationstudies, finished product test results, and dissolution profiles.

Many of the FDAs postapproval, premarketing inspections result in cita-tions because validation (and consistency) of the full-scale batches could not be es-tablished owing to problems with product dissolution, content uniformity, and po-tency. Validation reports on batch scale-ups may also reflect selective reporting ofdata. Of practical importance are the issues associated with a technology transferin a global market. Equipment standardization inevitably will cause a variety of en-gineering and process optimization concerns that can be classified as SUPAC.

This book presents the significant aspects of pharmaceutical scale-up to il-lustrate potential concerns, theoretical considerations, and practical solutions basedon the experience of the contributing authors. A prudent reader may use this hand-book as a reference and an initial resource for further study of the scale-up issues.

Michael Levin

REFERENCES

1. M Zlokarnik. Dimensional Analysis and Scale-Up in Chemical Engineering. NewYork: Springer-Verlag, 1991.

2. E Buckingham. On physically similar systems: Illustrations of the use of dimensionalequations. Phys Rev NY 4:345376, 1914.

3. M Zlokarnik. Problems in the application of dimensional analysis and scale-up of mix-ing operations. Chem Eng Sci 53(17):30233030, 1998.

4. H Leuenberger. Scale-up of granulation processes with reference to process monitor-ing. Acta Pharm Technol 29(4): 274280, 1983.

x Introduction

Contents

Preface vIntroduction vii

Contributors xv

1. Dimensional Analysis and Scale-Up in Theory and Industrial Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Marko Zlokarnik

2. Parenteral Drug Scale-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Igor Gorsky

3. Nonparenteral Liquids and Semisolids . . . . . . . . . . . . . . . . . . . . 57Lawrence H. Block

4. Scale-Up Considerations for Biotechnology-Derived Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Marco A. Cacciuttolo, Erica Shane, Roy Kimura, Cynthia Oliver,and Eric Tsao

5 (1). Batch Size Increase in Dry Blending and Mixing . . . . . . . . . . . 115Albert W. Alexander and Fernando J. Muzzio

5 (2). Powder Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133James K. Prescott

xi

6. Scale-Up in the Field of Granulation and Drying . . . . . . . . . . . . 151Hans Leuenberger

7. Batch Size Increase in Fluid Bed Granulation . . . . . . . . . . . . . . 171Dilip M. Parikh

8 (1). Scale-Up of the Compaction and Tableting Process . . . . . . . . . 221Joseph B. Schwartz

8 (2). Practical Aspects of Tableting Scale-Up . . . . . . . . . . . . . . . . . . 239Walter A. Strathy and Adolfo L. Gomez

8 (3). Dimensional Analysis of the Tableting Process . . . . . . . . . . . . . 253Michael Levin and Marko Zlokarnik

9. Scale-Up of Film Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Stuart C. Porter

10. Engineering Aspects of Process Scale-Up and Pilot Plant Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Adolfo L. Gomez and Walter A. Strathy

11. A Collaborative Search for Efficient Methods of Ensuring Unchanged Product Quality and Performance During Scale-Up of Immediate-Release Solid Oral Dosage Forms . . . . . . . . . . . . 325Ajaz S. Hussain

APPENDIXES: GUIDANCE FOR INDUSTRY

A. Immediate Release Solid Oral Dosage FormsScale-Up and Postapproval Changes: Chemistry, Manufacturing, and Controls, In Vitro Dissolution Testing, and In Vivo Bioequivalence Documentation . . . . . . . . . . . . . . . . . . . . . . . . . 353

B. SUPAC-MR: Modified Release Solid Oral Dosage FormsScale-Up and Postapproval Changes: Chemistry, Manufacturing, and Controls; In Vitro Dissolution Testing and In Vivo Bioequivalence Documentation . . . . . . . . . . . . . . . . . . . 373

C. SUPAC-IR/MR: Immediate Release and Modified Release Solid Oral Dosage FormsManufacturing Equipment Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

xii Contents

D. Extended Release Oral Dosage FormsDevelopment, Evaluation, and Application of In Vitro/In Vivo Correlations . . 447

E. Nonsterile Semisolid Dosage FormsScale-Up and Postapproval Changes: Chemistry, Manufacturing, and Controls; In Vitro Release Testing and In Vivo Bioequivalence Documentation SUPAC-SS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

F. SUPAC-SS: Nonsterile Semisolid Dosage FormsManufacturing Equipment Addendum . . . . . . . . . . . . . 499

G. Changes to an Approved NDA or ANDA . . . . . . . . . . . . . . . . . . 517

H. Waiver of In Vivo Bioavailability and Bioequivalence Studies for Immediate-Release Solid Oral Dosage Forms Based on a Biopharmaceutics Classification System . . . . . . . . . . . . . . . . . . 551

Index 565

Contents xiii

Contributors

Albert W. Alexander Department of Chemical and Biochemical Engineering,Rutgers University, Piscataway, New Jersey

Lawrence H. Block Division of Pharmaceutical Sciences, Duquesne Univer-sity, Pittsburgh, Pennsylvania

Marco A. Cacciuttolo Cell Culture Development, Biopharmaceutical Produc-tion, Medarex, Inc., Bloomsbury, New Jersey

Adolfo L. Gomez I.D.E.A.S., Inc., Wilson, North Carolina

Igor Gorsky Department of Technical Services, Alpharma, Baltimore, Mary-land

Ajaz S. Hussain Center for Drug Evaluation and Research, U.S. Food and DrugAdministration, Rockville, Maryland

Roy Kimura Onyx Pharmaceuticals, Inc., Richmond, California

Hans Leuenberger Institute of Pharmaceutical Technology, University ofBasel, Basel, Switzerland

Michael Levin Metropolitan Computing Corporation, East Hanover, New Jersey

xv

Fernando J. Muzzio Department of Chemical and Biochemical Engineering,Rutgers University, Piscataway, New Jersey

Cynthia Oliver Process Biochemistry, MedImmune, Inc., Gaithersburg, Mary-land

Dilip M. Parikh APACE Pharma Inc., Westminster, Maryland

Stuart C. Porter Pharmaceutical Technologies International, Inc., Belle Mead,New Jersey

James K. Prescott Jenike & Johanson, Inc., Westford, Massachusetts

Joseph B. Schwartz Philadelphia College of Pharmacy, Philadelphia, Pennsyl-vania

Erica Shane Process Biochemistry, MedImmune, Inc., Gaithersburg, Maryland

Walter A. Strathy I.D.E.A.S., Inc., Wilson, North Carolina

Eric Tsao Process Cell Culture, MedImmune, Inc., Gaithersburg, Maryland

Marko Zlokarnik Graz, Austria

xvi Contributors

1Dimensional Analysis and Scale-Upin Theory and Industrial Application

Marko ZlokarnikGraz, Austria

I. INTRODUCTION

A chemical engineer is generally concerned with the industrial implementation ofprocesses in which chemical or microbiological conversion of material takes placein conjunction with the transfer of mass, heat, and momentum. These processes arescale dependent; that is, they behave differently on a small scale (in laboratories orpilot plants) and on a large scale (in production). They include heterogeneouschemical reactions and most unit operations. Understandably, chemical engineershave always wanted to find ways of simulating these processes in models to gaininsights that will assist them in designing new industrial plants. Occasionally, theyare faced with the same problem for another reason: An industrial facility alreadyexists but will not function properly, if at all, and suitable measurements have to becarried out to discover the cause of the difficulties and provide a solution.

Irrespective of whether the model involved represents a scale-up or ascale-down, certain important questions always apply:

1. How small can the model be? Is one model sufficient, or should tests becarried out in models of different sizes?

2. When must or when can physical properties differ? When must themeasurements be carried out on the model with the original system ofmaterials?

3. Which rules govern the adaptation of the process parameters in themodel measurements to those of the full-scale plant?

4. Is it possible to achieve complete similarity between the processes inthe model and those in its full-scale counterpart? If not, how should oneproceed?

1

These questions touch on the fundamentals of the theory of models, whichare based on dimensional analysis. Although they have been used in the field offluid dynamics and heat transfer for more than a centurycars, aircraft, vessels,and heat exchangers were scaled up according to these principlesthese methodshave gained only a modest acceptance in chemical engineering. University gradu-ates are usually not skilled enough to deal with such problems at all. On the otherhand, there is no motivation for this type of research at universities, since, as a rule,they are not confronted with scale-up tasks and are not equipped with the necessaryapparatus on the bench scale. All this gives a totally wrong impression that thesemethods are, at most, of marginal importance in practical chemical engineering, forotherwise would they have been taught and dealt with in greater depth.

II. DIMENSIONAL ANALYSISA. The Fundamental PrincipleDimensional analysis is based upon the recognition that a mathematical formula-tion of a physicotechnological problem can be of general validity only when theprocess equation is dimensionally homogenous, which means that it must be validin any system of dimensions.

B. What Is a Dimension?A dimension is a purely qualitative description of a perception of a physical entityor a natural appearance. A length can be experienced as a height, a depth, abreadth. A mass presents itself as a light or heavy body, time as a short momentor a long period. The dimension of a length is length (L), the dimension of a massis mass (M), etc.

C. What Is a Physical Quantity?Unlike a dimension, a physical quantity represents a quantitative description of aphysical quality (e.g., a mass of 5 kg). It consists of a measuring unit and a nu-merical value. The measuring unit of length can be a meter, a foot, a cubit, a yard-stick, a nautical mile, a light year, etc. The measuring units of energy are, e.g.,joules, cal, eV. (It is therefore necessary to establish the measuring units in an ap-propriate measuring system.)

D. Basic and Derived Quantities, Dimensional ConstantsA distinction is being made between basic and secondary quantities, the latter of-ten being referred to as derived quantities. Basic quantities are based on standardsand are quantified by comparison with them. Secondary units are derived from the

2 Zlokarnik

primary ones according to physical laws, e.g., velocity length/ time. (The bor-derline separating both types of quantities is largely arbitrary: 50 years ago a mea-suring system was used in which force was a primary dimension instead of mass!)

All secondary units must be coherent with the basic units (Table 1); e.g., themeasuring unit of velocity must not be miles/hr or km/hr but meters/sec!

If a secondary unit has been established by a physical law, it can happen thatit contradicts another one. Example: According to the Newtons second law ofmotion, the force F is expressed as a product of mass m and acceleration a:F ma, having the measuring unit of [kgm/sec2 N]. According to the New-tons law of gravitation, force is defined by F m1m2 /r2, thus leading to a com-pletely different measuring unit [kg2/m2]. To remedy this, the gravitational con-stant Ga dimensional constanthad to be introduced to ensure the dimensionalhomogeneity of the latter equation: F Gm1m2 /r2. Another example affects theuniversal gas constant R, the introduction of which ensures that in the perfect gasequation of state pV nRT, the secondary unit for work W pV [ML2T2] is notoffended.

Another class of derived quantities is represented by the coefficients in di-verse physical equations, e.g., transfer equations. They are established by the re-spective equations and determined via measurement of their constituents, e.g.,heat and mass transfer coefficients.

E. Dimensional SystemsA dimensional system consists of all the primary and secondary dimensions andcorresponding measuring units. The currently used International System of Di-mensions (Systme International dunits, SI) is based on seven basic dimensions.They are presented in Table 1 together with their corresponding basic units. Forsome of them a few explanatory remarks may be necessary.

Temperature expresses the thermal level of a system and not its energeticcontents. (A fivefold mass of a matter has the fivefold thermal energy at the sametemperature!) The thermal energy of a system can indeed be converted into me-

Dimensional Analysis 3

Table 1 Base Quantities, Their Dimensions, and Their Units According to SIBase quantity Base dimension Base unit

Length L m (meter)Mass M kg (kilogram)Time T sec (second)Thermodynamic temperature K (Kelvin)Amount of substance N mol (mole)Electric current I A (ampere)Luminous intensity Iv cd (candela)

chanical energy (base unit, joule). Moles are the amount of matter and must not beconfused with the quantity of mass. The molecules react as individual entities re-gardless of their mass: One mole of hydrogen (2 g/mol) reacts with one mole ofchlorine (71 g/mol) to produce two moles of hydrochloric acid, HCl. Table 2shows the most important secondary dimensions. Table 3 refers to some very fre-quently used secondary units that have been named after famous researchers.

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Table 2 Often-Used Physical Quantities and TheirDimensions According to the Currently Used SI inMechanical and Thermal Problems

Physical quantity Dimension

Angular velocity T1Shear rate, frequencyMass transfer coefficient kLaVelocity L T1Acceleration L T2Kinematic viscosity L2 T1Diffusion coefficientThermal diffusivityDensity M L3Surface tension M T2Dynamic viscosity M L1 T1Momentum M L T1Force M L T2Pressure, stress M L1 T2Angular momentum M L2 T1Energy, work, torque M L2 T2Power M L2 T3Heat capacity L2 T2 1Thermal conductivity M L T3 1Heat transfer coefficient M T3 1

Table 3 Important Secondary Measuring Units in Mechanics, Named After FamousResearchers

Secondary Abbreviationquantity Dimension Measuring unit for:

Force M L T2 kg m sec2 (N) NewtonPressure M L1 T2 kg m1 sec2 (Pa) PascalEnergy M L2 T2 kg m2 sec2 (J) JoulePower M L2 T3 kg m2 sec3 (W) Watt

F. Dimensional Homogeneity of a Physical ContentThe aim of dimensional analysis is to check whether or not the physical contentunder examination can be formulated in a dimensionally homogeneous manner.The procedure necessary to accomplish this consists of two parts:

1. First, all physical parameters necessary to describe the problem arelisted. This so-called relevance list of the problem consists of thequantity in question and of all the parameters that influence it. In eachcase only one target quantity must be considered; it is the only depen-dent variable. On the other hand, all the influencing parameters must beprimarily independent of each other.

2. In the second step the dimensional homogeneity of the physical contentis checked by transferring it into a dimensionless form. Note: A physi-cal content that can be transformed into dimensionless expressions isdimensionally homogeneous!

The information given to this point will be made clear by the followingamusing but instructive example.Example 1: What Is the Correlation Between the Baking Time and the Weight ofa Christmas Turkey? We first recall the physical situation. To facilitate this wedraw a sketch (Sketch 1). At high oven temperatures the heat is transferred fromthe heating elements to the meat surface by both radiation and heat convection.From there it is transferred solely by the unsteady-state heat conduction that surelyrepresents the rate-limiting step of the whole heating process.

Physical quantity Symbol Dimension

Baking time TSurface of meat A L2Thermal diffusivity a L2 T1Temperature on the surface T0 Temperature distribution T

The higher the thermal conductivity of the body, the faster the heatspreads out. The higher its volume-related heat capacity Cp, the slower the heattransfer. Therefore, the unsteady-state heat conduction is characterized by onlyone material property, the thermal diffusivity a /Cp of the body.

Baking is an endothermal process. The meat is cooked when a certain tem-perature distribution (T) is reached. Its about the time necessary to achieve thistemperature range.

After these considerations we are able to precisely construct the relevance list:

{, A, a, T0, T } (1)


The base dimension of temperature appears in only two parameters. They cantherefore produce only one dimensionless quantity:

1 T/ T0 or (T0 T) / T0 (2)The three residual quantities form one additional dimensionless number:

2 a/A Fo (3)In the theory of heat transfer, 2 is known as the Fourier number. Therefore, thebaking procedure can be presented in a two-dimensional frame:

T/T0 (Fo) (4)Here, five dimensional quantities [Eq. (1)] produce two dimensionless numbers[Eq. (4)]. This had to be expected because the dimensions in question are made upof three basic dimensions: 5 3 2 (see pi theorem).

We can now easily answer the question concerning the correlation betweenthe baking time and the weight of the Christmas turkey, without explicitly know-ing the function that connects both numbers, Eq. (4). To achieve the same tem-perature distribution T /T0 or (T0 T) /T0 in different-size bodies, the dimension-less quantity a/A Fo must have the same ( idem) value. Due to the fact thatthe thermal diffusivity a remains unaltered in meat of the same kind (a idem),this demand leads to

T/T0 idem Fo a/A idem /A idem A (5)This statement is obviously useless as a scale-up rule because meat is bought ac-cording to weight and not surface. We can remedy this simply: In bodies, the fol-lowing correlation between mass m, surface A, and volume V exists:

m V L3 A32 (A L2) (6)

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Sketch 1


Table 4 Important Named Dimensionless Numbers

Name Symbol Group Remarks

A. Mechanical unit operationsReynolds Re v1/ /Froude Fr v2/(1g)

Fr* v2/(1g) Fr(/)Galilei Ga g13/2 Re2/FrArchmedes Ar g 13/2 Ga (/)Euler Eu p/(v2)Newton Ne F/(v212) force

P/(v312) powerWeber We v21/Ohnesorg Oh /(1)12 We 12/ReMach Ma v/vs vs velocity of soundKnudsen Kn m/1 m molecular free path lengthB. Thermal unit operations (heat transfer)Nusselt Nu h1/Prandtl Pr /a a /(Cp)Grashof Gr Tg13/2 TGaFourier Fo at/12Pclet Pe v1/a RePrRayleigh Ra Tg13/(a) GrPrStanton St h/(vCp) Nu/(RePr)C. Thermal unit operations (mass transfer)Sherwood Sh k1/D k mass transfer coefficientSchmidt Sc /DBodenstein Bo v1/Dax Dax axial dispersion coefficientLewis Le a/D Sc/PrStanton St k/v Sh/(Re Sc)D. Chemical reaction engineeringArrhenius Arr E/(RT) E activation energyHatta Hat1 (k1D)12/kL 1st-order reaction

Hat2 (k2c2D)12/kL 2nd-order reactionDamkhler Da

cCHpT

R

0 Genuine; see Ref. 5

DaI k1 residence timeDaII k112/D DaI BoDaIII k1 CcpHTR0 DaI cCHpTR0DaIV

k1cHT

R

0

12 DaI Re Pr cCHpTR0

Therefore, from idem it follows that

A m23

and by this

A m23 2 /1 (m2 /m1)

23 (7)

This is the scale-up rule for baking or cooking time in the case of meat of the samekind (a, idem). It states that by doubling the mass of meat, the cooking timewill increase by 2

23 1.58.

G. B. West [1] refers to (inferior) cookbooks that simply say something like20 minutes per pound, implying a linear relationship with weight. However, su-perior cookbooks exist such as the Better Homes and Gardens Cookbook (DesMoines Meredith Corp., 1962), that recognize the nonlinear nature of this rela-tionship. The graphical representation of measurements in this book confirms therelationship

m0.6 (8)which is very close to the theoretical evaluation giving m

23 m0.67.

The elegant solution of this first example should not tempt the reader tobelieve that dimensional analysis can be used to solve every problem. To treatthis example by dimensional analysis, the physics of unsteady-state heat con-duction had to be understood. Bridgmans [2] comment on this situation is par-ticularly appropriate: The problem cannot be solved by the philosopher in hisarmchair, but the knowledge involved was gathered only by someone at sometime soiling his hands with direct contact. This transparent and easy exampleclearly shows how dimensional analysis deals with specific problems and whatconclusions it allows. It should now be easier to understand Lord Rayleighs sar-castic comment with which he began his short essay on The Principle of Simil-itude [3]: I have often been impressed by the scanty attention paid even byoriginal workers in physics to the great principle of similitude. It happens not in-frequently that results in the form of laws are put forward as novelties on thebasis of elaborate experiments, which might have been predicted a priori after afew minutes consideration.

From the foregoing example we also learn that a transformation of a physi-cal dependency from a dimensional into a dimensionless form is automatically ac-companied by an essential compression of the statement: The set of the dimen-sionless numbers is smaller than the set of the quantities contained in them, but itdescribes the problem equally comprehensively. In our example the dependencybetween five dimensional parameters is reduced to a dependency between onlytwo dimensionless numbers! This is the proof of the so-called pi theorem (pi after, the sign used for products).

8 Zlokarnik

G. The Pi Theorem

Every physical relationship between n physical quantities can be reduced to arelationship between m n r mutually independent dimensionless groups,whereby r stands for the rank of the dimensional matrix, made up of the phys-ical quantities in question and generally equal to the number of the basicquantities contained in them.

(The pi theorem is often associated with the name of E. Buckingham [4], becausehe introduced this term in 1914. But the proof of it had already been accomplishedin the course of a mathematical analysis of partial differential equations by A. Fe-dermann in 1911; see Ref. 5.)

III. THE DETERMINATION OF A PI SET BY MATRIXCALCULATION

A. The Establishment of a Relevance List of a ProblemAs a rule, more than two dimensionless numbers will be necessary to describe aphysicotechnological problem, and therefore they cannot be derived by themethod just described. In this case the easy and transparent matrix calculation in-troduced by J. Pawlowski [6] is increasingly used. It will be demonstrated by thefollowing example. It treats an important problem in industrial chemistry andbiotechnology, because the contact between gas and liquid in mixing vessels oc-curs very frequently in mixing operations.Example 2: The Determination of the Pi Set for the Stirrer Power in the ContactBetween Gas and Liquid. We examine the power consumption of a turbine stir-rer (so-called Rushton turbine; see inset in Fig. 1) installed in a baffled vessel andsupplied by gas from below (see Sketch 2). We facilitate the procedure by sys-tematically listing the target quantity and all the parameters influencing it:

1. Target quantity: mixing power P2. Influencing parameters

a. Geometrical: stirrer diameter db. Physical properties:

Fluid density Kinematic viscosity

c. Process related:Stirrer speed nGas throughput qGravitational acceleration g

The relevance list is:

{P; d; , ; n, q, g} (9)


We interrupt the procedure to ask some important questions concerning: (1)the determination of the characteristic geometric parameter, (2) the setting of allrelevant material properties, and (3) the taking into account the gravitationalacceleration.

1. Determination of the characteristic geometric parameter: It is obviousthat we could name all the geometric parameters indicated in Sketch 2. They wereall the geometric parameters of the stirrer and of the vessel, especially its diame-ter D and the liquid height H. In case of complex geometry such a procedurewould necessarily deflect us from the problem. It is therefore advisable to intro-duce only one characteristic geometric parameter, knowing that all the others canbe transformed into dimensionless geometric numbers by division with this one.As the characteristic geometric parameter in the Example 2, the stirrer diameterwas introduced. This is reasonable. One can imagine how the mixing power wouldreact to an increase in the vessel diameter D: It is obvious that from a certain Don, it would have no influence, but a small change of the stirrer diameter d wouldalways have an impact!

2. Setting of all relevant material properties: In the preceding relevancelist, only the density and the viscosity of the liquid were introduced. The materialproperties of the gas are of no importance as compared with the physical propertiesof the liquid. It was also ascertained by measurements that the interfacial tension does not effect the stirrer power. Furthermore, measurements [7] revealed that thecoalescence behavior of the material system is not affected if aqueous glycerol orcane syrup mixtures are used to increase viscosity in model experiments.

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Sketch 2

3. The importance of the gravitational constant: Due to the extreme den-sity difference between gas and liquid (ca. 1:1.000), it must be expected that thegravitational acceleration g will exert a big influence. One should actually writeg, butsince L G Lthe dimensionless number would contain g /L gL /L g.

We now proceed to solve Example 2.

B. Constructing and Solving the Dimensional MatrixIn transforming the relevance listEq. (9)of the preceding seven physicalquantities into a dimensional matrix, the following should be kept in mind in or-der to minimize the calculations required.

1. The dimensional matrix consists of a square core matrix and a residualmatrix.

2. The rows of the matrix are formed of base dimensions, contained in thedimensions of the quantities, and they will determine the rank r of thematrix. The columns of the matrix represent the physical quantities orparameters.

3. Quantities of the square core matrix may eventually appear in all of thedimensionless numbers as fillers, whereas each element of the resid-ual matrix will appear in only one dimensionless number. For this rea-son the residual matrix should be loaded with essential variables likethe target quantity and the most important physical properties and pro-cess-related parameters.

4. By theextremely easy!matrix rearrangement (linear transforma-tions), the core matrix is transformed into a matrix of unity. The maindiagonal consists only of ones and the remaining elements are all zero.One should therefore arrange the quantities in the core matrix in a wayto facilitate this procedure.

5. After the generation of the matrix of unity, the dimensionless numbersare created as follows: Each element of the residual matrix forms thenumerator of a fraction, while its denominator consists of the fillersfrom the matrix of unity with the exponents indicated in the residualmatrix.

Let us return to Example 2. The dimensional matrix reads:

d n P q g

Mass M 1 0 0 1 0 0 0Length L 3 1 0 2 2 3 1Time T 0 0 1 3 1 1 2

core matrix residual matrix


Only one linear transformation is necessary to transform 3 in L-row/-columninto zero. The subsequent multiplication of the T-row by 1 transfers 1 to 1:

d n P q g

M 1 0 0 1 0 0 03M L 0 1 0 5 2 3 1

T 0 0 1 3 1 1 2unity matrix residual matrix

The residual matrix contains four parameters, so four numbers result:

1n

P3d5

nP3d5 Ne (Newton number)

0n

1d2

n

d2 Re1 (Reynolds number)

dq3n Q (Gas throughput number)

dgn2 Fr1 (Froude number)

The interdependence of seven dimensional quantities of the relevance list, Eq. (9),reduces to a set of only 7 3 4 dimensionless numbers:

{Ne, Re, Q, Fr} or (Ne, Re, Q, Fr) 0 (10)thus again confirming the pi theorem.

C. Determination of the Process CharacteristicsThe functional dependency, Eq. (10), is the maximum that dimensional analysiscan offer here. It cannot provide any information about the form of the function .This can be accomplished solely by experiments.

The first question we must ask is: Are laboratory tests, performed in onesingle piece of laboratory apparatusi.e., on one single scalecapable of pro-viding binding information on the decisive process number? The answer here isaffirmative. We can change Fr by means of the rotational speed of the stirrer, Qby means of the gas throughput, and Re by means of the liquid viscosity inde-pendent of each other.

The results of these model experiments are described in detail in Ref. 7. Forour consideration, it is sufficient to present only the main result here. This statesthat, in the industrially interesting range (Re 104 and Fr 0.65), the power

12 Zlokarnik

number Ne is dependent only on the gas throughput number Q; see Figure 1. Byraising the gas throughput number Q and thus enhancing gas hold-up in the liquid,liquid density diminishes and the Newton number Ne decreases to only one-thirdof its value in a nongassed liquid.

Knowledge of this power characteristic, the analytical expression forwhich is

Ne 1.5 (0.5Q0.075 1600Q2.6)1 (Q 0.15) (11)can be used to reliably design a stirrer drive for the performance of material con-versions in the gas/ liquid system (e.g., oxidations with O2 or air, fermentations)as long as the physical, geometric, and process-related boundary conditions (Re,Fr, and Q) comply with those of the model measurement.

IV. FUNDAMENTALS OF THE THEORY OF MODELS ANDOF SCALE-UP

A. Theory of ModelsThe results in Figure 1 have been obtained by changing the rotational speed ofthe stirrer and the gas throughput, whereas the liquid properties and the charac-teristic length (stirrer diameter d ) remained constant. But these results couldhave also been obtained by changing the stirrer diameter. It does not matter bywhich means a relevant number (here Q) is changed because it is dimensionlessand therefore independent of scale (scale-invariant). This fact presents the ba-


Figure 1 Power characteristics of a turbine stirrer (Rushton turbine) in the range Re 104 and Fr 0.65 for two D/d values. Material system: water/air. (From Ref. 7.)

sis for a reliable scale-up:

Two processes may be considered completely similar if they take place insimilar geometrical space and if all the dimensionless numbers necessary todescribe them have the same numerical value (i identical or idem).

Clearly, the scale-up of a desired process condition from a model to industrialscale can be accomplished reliably only if the problem was formulated and dealtwith according to dimensional analysis!

B. Model Experiments and Scale-UpIn the foregoing example the process characteristics (here power characteristics)presenting a comprehensive description of the process were evaluated. This oftenexpensive and time-consuming method is certainly not necessary if one only hasto scale-up a given process condition from the model to the industrial plant (orvice versa). With the last example, and assuming that the Ne(Q) characteristic likethat in Figure 1 is not explicitly known, the task is to predict the power consump-tion of a Rushton turbine of d 0.8 m, installed in a baffled vessel of D 4 m(D/d 5) and rotating with n 200 min1. The air throughput is q 500 m3/hrand the material system is water/air.

One only needs to knowand this is essentialthat the hydrodynamics inthis case are governed solely by the gas throughput number and that the process isdescribed by an unknown dependency Ne(Q). Then one can calculate the Q num-ber of the industrial plant:

Q q/nd3 8.14 102

What will the power consumption of the turbine be?Let us assume that we have a geometrically similar laboratory device of D

0.4 m (V 0.050 m3) with the turbine stirrer of d 0.08 m and that the rota-tional speed of the stirrer is n 750 min1. What must the gas throughput be toobtain Q idem in the laboratory device? The answer is

q/nd3 8.14 102 q 1.88 m3/hr

Under these conditions the stirrer power must be measured and the power numberNe P/(n3d5) calculated. We find Ne 1.75. Due to the fact that Q idem re-sults in Ne idem, the power consumption PT of the industrial stirrer can be ob-tained:

Ne idem NeT NeM nP3d5T nP3d5M (12)From Ne 1.75 found in laboratory measurement, the power P of the industrialturbine stirrer of d 0.8 m and a rotational speed of n 200 min1 is calculated

14 Zlokarnik

as follows:

P Nen3d5 1.75 1 103 (200/60)3 0.85 21,200 W 21 kWThis results in 21/50 kW/m3 0.42 kW/m3, which is a fair volume-related powerinput for many conversions in the gas/liquid system.

We realize that in scale-up, comprehensive knowledge of the functional de-pendency (i) 0like that in Figure 1is not necessary. All we need is toknow which pi space describes the process.

V. FURTHER PROCEDURES TO ESTABLISH ARELEVANCE LIST

A. Consideration of the Acceleration Due to Gravity gIf a natural or universal physical constant has an impact on the process, it has tobe incorporated into the relevance list, whether it will be altered or not. In this con-text the greatest mistakes are made with regard to the gravitational constant g.Lord Rayleigh [3] complained bitterly, saying: I refer to the manner in whichgravity is treated. When the question under consideration depends essentiallyupon gravity, the symbol of gravity (g) makes no appearance, but when gravitydoes not enter the question at all, g obtrudes itself conspicuously. This is all themore surprising in view of the fact that the relevance of this quantity is easyenough to recognize if one asks the following question: Would the process func-tion differently if it took place on the moon instead of on Earth? If the answer tothis question is affirmative, g is a relevant variable.

The gravitational acceleration g can be effective solely in connection withdensity as gravity g. When inertial forces play a role, the density has to be listedadditionally. Thus it follows that:

1. In cases involving the ballistic movement of bodies, the formation ofvortices in stirring, the bow wave of a ship, the movement of a pendu-lum, and oher processes affected by the Earths gravity, the relevancelist comprises g and .

2. Creeping flow in a gravitational field is governed by the gravity galone.

3. In heterogeneous physical systems with density differences (sedimen-tation or buoyancy), the gravity difference g and play a decisiverole.

In the second example we already treated a problem where the gravitationalconstant is of prime importance, due to extreme difference in densities in thegas/liquid system, provided that the Froude number is low: Fr 0.65.


B. Introduction of Intermediate QuantitiesMany engineering problems involve several parameters that impede the

elaboration of the pi space. Fortunately, in some cases a closer look at a problem(or previous experience) facilitates reduction of the number of physical quantitiesin the relevance list. This is the case when some relevant variables affect the pro-cess by way of a so-called intermediate quantity. Assuming that this intermedi-ate variable can be measured experimentally, it should be included in the problemrelevance list if this facilitates the removal of more than one variable from the list.

The fluid velocity v in pipesor the superficial gas velocity vG in mixingvessels or in bubble columnspresents a well-known intermediate quantity. Itsintroduction into the relevance list removes two others (throughput q and diame-ter D), because v q/D2 and vG qG/D2, respectively.

The impact, which the introduction of intermediate quantities can have onthe relevance list, will be demonstrated in the following by one elegant example.Example 3: Mixing-Time Characteristics for Liquid Mixtures with Differences inDensity and Viscosity. The mixing time necessary to achieve a molecular ho-mogeneity of a liquid mixturenormally measured by decolorization methodsde-pends, in material systems without differences in density and viscosity, on only fourparameters: stirrer diameter d, density , kinematic viscosity , rotational speed n:

{; d; , ; n} (13)From this, the mixing-time characteristics are

n (Re) Re nd2/ (14)See Example 5.2 later and Figure 10.

In material systems with differences in density and viscosity, the relevancelist, Eq. (13), enlarges by the physical properties of the second mixing component,by the volume ratio of both phases V2 /V1, and, due to the density differences,inevitably by the gravity difference g to nine parameters:

{; d; 1, 1, 2, 2, ; g, n} (15)This results in a mixing-time characteristics incorporating six numbers:

n (Re, Ar, 2/1, 2/1, ) (16)Re nd2/1 Reynolds number,

Ar g d3/(121) Archimedes numberMeticulous observation of this mixing process (the slow disappearance of

the Schlieren patterns as result of the disappearance of density differences) revealsthat macromixing is quickly accomplished compared to the micromixing. Thistime-consuming process already takes place in a material system that can be fully

16 Zlokarnik

described by the physical properties of the mixture:

* (1, 2, ) and * (1, 2, ) (17)By introducing these intermediate quantities * and *, the nine-parameter rele-vance list, Eq. (15), reduces by three parameters to a six-parametric one:

{; d; *, *; g, n} (18)and gives a mixing characteristics of only three numbers:

n (Re, Ar) (19)(In this case, Re and Ar have to be formed by * and *!)

The process characteristics of a cross-beam stirrer was established in this pispace by evaluation of corresponding measurements in two different-size mixingvessels (D 0.3 and 0.6 m) using different liquid mixtures (/* 0.01 0.29and 2/1 1 5300). It reads [8]:

n 51.6 Re1(Ar1/3 3) Re 101 105; Ar 102 1011 (20)This example clearly shows the big advantages achieved by the introduction

of intermediate quantities. This will also be made clear by upcoming Example 4.

C. Material Systems of Unknown Physical PropertiesWith the foams, sludges, and slimes often encountered in biotechnology, we areconfronted with the problem of not being able to list the physical properties be-cause they are still unknown and therefore cannot be quantified. This situation of-ten leads to the opinion that dimensional analysis would fail in such cases.

It is obvious that this conclusion is wrong: Dimensional analysis is a methodbased on logical and mathematical fundamentals [2,6]. If relevant parameters can-not be listed because they are unknown, one cannot blame the method! The onlysolution is to perform the model measurements with the same material system andto change the model scales.Example 4: Scale-Up of a Mechanical Foam Breaker. The question is posedabout the mode of performing and evaluating model measurements with a giventype of mechanical foam breaker (foam centrifuge; see sketch in Fig. 2) to obtainreliable information on dimensioning and scale-up of these devices. Preliminaryexperiments have shown that for each foam emergenceproportional to the gasthroughput qGfor each foam breaker of diameter d, a minimum rotational speednmin exists that is necessary to control it. The dynamic properties of the foam (e.g.,density and viscosity, elasticity of the foam lamella) cannot be fully named ormeasured. We will have to content ourselves with listing them wholesale as ma-terial properties Si. In our model experiments we will of course be able to replaceSi by the known type of surfactant (foamer) and its concentration c [ppm].


In discerning the process parameters we realize that the gravitational accel-eration g has no impact on the foam breaking within the foam centrifuge: The cen-trifugal acceleration n2d exceeds the gravitational one (g) by far! However, wehave to recognize that the water content of the foam entering the centrifuge de-pends very much on the gravitational acceleration: On the moon the waterdrainage would be by far less effective! In contrast to the dimensional analysispresented in Ref. 9, we are well advised to add g to the relevance list:

{nmin; d; type of foamer, c; qG, g} (21)For the sake of simplicity, in the following nmin will be replaced by n and qG

by q. For each type of foamer we obtain the following pi space:

nqd3

, ng2d, c or, abbreviated, {Q1, Fr, c} (22)

To prove this pi space, measurements in different-size model equipment are nec-essary to produce reliable process characteristics. For a particular foamer (Merso-lat H of Bayer AG, Germany) the results are given in Figure 2. They fully confirmthe pi space [Eq. (22)].

The straight line in Figure 2 corresponds to the analytic expression

Q1 Fr0.4c 0.32 (23)

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Figure 2 Process characteristics of the foam centrifuge (sketch) for a particular foamer(Mersolat H of Bayer AG, Germany). (From Ref. 9.)

which reduces to

nd const q0.2(c) (24)Here, the foam breaker will be scaled up according to its tip speed u nd inmodel experiments, which will also depend moderately on the foam yield (q).

In all other foamers examined [9], the correspondence Q1 Fr0.45 wasfound. If the correlation

Q1 Fr0.5(c) (25)proves to be true, then it can be reduced to

n2d/g const(c) (26)In this case the centrifugal acceleration (n2d) would present the scale-up criterionand would depend only on the foamer concentration and not on foam yield (q).

D. Short Summary of the Essentials of DimensionalAnalysis and Scale-Up

The advantages made possible by correct and timely use of dimensional analysisare as follows.

1. Reduction of the number of parameters required to define the problem.The pi theorem states that a physical problem can always be describedin dimensionless terms. This has the advantage that the number of di-mensionless groups that fully describe it is much smaller than the num-ber of dimensional physical quantities. It is generally equal to the num-ber of physical quantities minus the number of base units contained inthem.

2. Reliable scale-up of the desired operating conditions from the model tothe full-scale plant. According to the theory of models, two processesmay be considered similar to one another if they take place under geo-metrically similar conditions and all dimensionless numbers which de-scribe the process have the same numerical value.

3. A deeper insight into the physical nature of the process. By presentingexperimental data in a dimensionless form, one distinct physical statecan be isolated from another (e.g., turbulent or laminar flow region) andthe effect of individual physical variables can be identified.

4. Flexibility in the choice of parameters and their reliable extrapolationwithin the range covered by the dimensionless numbers. These advan-tages become clear if one considers the well-known Reynolds number,Re vL/, which can be varied by altering the characteristic velocityv or a characteristic length L or the kinematic viscosity . By choosing


appropriate model fluids, the viscosity can very easily be altered by sev-eral orders of magnitude. Once the effect of the Reynolds number isknown, extrapolation of both v and L is allowed within the examinedrange of Re.

E. Area of Applicability of the Dimensional AnalysisThe application of dimensional analysis is indeed heavily dependent on the avail-able knowledge. The following five steps (see Fig. 3) can be outlined as:

1. The physics of the basic phenomenon is unknownDimensional anal-ysis cannot be applied.

2. Enough is known about the physics of the basic phenomenon to com-pile a first, tentative relevance listThe resultant pi set is unreliable.

3. All the relevant physical variables describing the problem are knownThe application of dimensional analysis is unproblematic.

4. The problem can be expressed in terms of a mathematical equationA

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Figure 3 Graphical representation of the four levels of knowledge and their impact onthe treatment of the problem by dimensional analysis. (From J. Pawlowski, personal com-munication, 1984.)

closer insight into the pi relationship is feasible and may facilitate a re-duction of the set of dimensionless numbers.

5. A mathematical solution of the problem existsThe application of di-mensional analysis is superfluous.

It must, of course, be said that approaching a problem from the point of viewof dimensional analysis also remains useful even if all the variables relevant to theproblem are not yet known: The timely application of dimensional analysis mayoften lead to the discovery of forgotten variables or the exclusion of artifacts.

F. Experimental Methods for Scale-UpIn Section I, a number of questions were posed that are often asked in connectionwith model experiments.

How small can a model be? The size of a model depends on the scale factorLT /LM and on the experimental precision of measurement. Where LT /LM 10, a10% margin of error may already be excessive. A larger scale for the model willtherefore have to be chosen to reduce the error.

Is one model scale sufficient, or should tests be carried out in models of dif-ferent sizes? One model scale is sufficient if the relevant numerical values of thedimensionless numbers necessary to describe the problem (the so-called processpoint in the pi space describing the operational condition of the technical plant)can be adjusted by choosing the appropriate process parameters or physical prop-erties of the model material system. If this is not possible, the process character-istics must be determined in models of different sizes, or the process point mustbe extrapolated from experiments in technical plants of different sizes.

When must model experiments be carried out exclusively with the originalmaterial system? Where the material model system is unavailable (e.g., in the caseof non-Newtonian fluids) or where the relevant physical properties are unknown(e.g., foams, sludges, slimes), the model experiments must be carried out with theoriginal material system. In this case measurements must be performed in modelsof various sizes (cf. Example 4).

G. Partial SimilarityThe theory of models requires that in the scale-up from a model (index M) to theindustrial scale (index T) not only the geometric similarity be ensured but also alldimensionless numbers describing the problem retain the same numerical values(i idem). This means, e.g., that in scale-up of boats or ships the dimensionlessnumbers governing the hydrodynamics here

Fr Lvg2 and Re vv

L


must retain their numerical values: FrT FrM and ReT ReM. It can easily beshown that this requirement cannot be fulfilled here!

Due to the fact that the gravitational acceleration g cannot be varied onEarth, the Froude number Fr of the model can be adjusted to that of the full-scalevessel only by its velocity vM. Subsequently, Re idem can be achieved only byadjusting the viscosity of the model fluid. In the case where the model size is only10% of the full size (scale factor LT /LM 10), Fr idem is achieved in the modelat vM 0.32vT. To fulfill Re idem, for the kinematic viscosity of the modelfluid vM it follows:

M

T

vv

M

T

LL

M

T 0.32 0.1 0.032

No liquid exists whose viscosity would be only 3% of that of water!We have to realize that sometimes requirements concerning physical prop-

erties of model materials exist that cannot be implemented. In such cases only apartial similarity can be realized. For this, essentially only two procedures areavailable (for details see Refs. 5 and 10). One consists of a well-planned experi-mental strategy, in which the process is divided into parts that are then investi-gated separately under conditions of complete similarity. This approach was firstapplied by William Froude (18101879) in his efforts to scale up the drag resis-tance of the ships hull.

The second approach consists in deliberately abandoning certain similaritycriteria and checking the effect on the entire process. This technique was used byGerhard Damkhler (19081944) in his trials to treat a chemical reaction in a cat-alytic fixed-bed reactor by means of dimensional analysis. Here the problem of asimultaneous mass and heat transfer arisesthey are two processes that obeycompletely different fundamental principles!

It is seldom realized that many rules of thumb utilized for scale-up of dif-ferent types of equipment are represented by quantities that fulfill only a partialsimilarity. As examples, only the volume-related mixing power P/Vwidely usedfor scaling-up mixing vesselsand the superficial velocity v, which is normallyused for scale-up of bubble columns, should be mentioned here.

The volume-related mixing power P/V presents an adequate scale-up crite-rion only in liquid/liquid dispersion processes and can be deduced from the perti-nent process characteristics dp/d We0.6 (dp is the particle or droplet diameter;We is the Weber number). In the most common mixing operation, the homoge-nization of miscible liquids, where a macro- and back-mixing is required, this cri-terion fails completely [10]!

Similarly, the superficial velocity v or vG of the gas throughput as an inten-sity quantity is a reliable scale-up criterion only in mass transfer in gas/liquid sys-tems in bubble columns. In mixing operations in bubble columns, requiring that

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the whole liquid content be back-mixed (e.g., in homogenization), this criterioncompletely loses its validity [10].

We have to draw the following conclusion: A particular scale-up criterionthat is valid in a given type of apparatus for a particular process is not necessarilyapplicable to other processes occurring in the same device.

VI. TREATMENT OF VARIABLE PHYSICAL PROPERTIESBY DIMENSIONAL ANALYSIS

It is generally assumed that the physical properties of the material system remainunaltered in the course of the process. Process equations, e.g., the heat character-istics of a mixing vessel or a smooth straight pipe

Nu (Re, Pr) (27)are valid for any material system with Newtonian viscosity and for any constantprocess temperature, i.e., for any constant physical property.

However, constancy of physical properties cannot be assumed in everyphysical process. A temperature field may well generate a viscosity field or evena density field in the material system treated. In non-Newtonian (pseudoplastic orviscoelastic) liquids, a shear rate can also produce a viscosity field.

In carrying out a scale-up, the industrial process has to be similar to the lab-oratory process in every relation. Besides the geometric and process-related sim-ilarity, it is self-evident that the fluid dynamics of the material system also has tobehave similarly. This requirement normally represents no problems when New-tonian fluid are treated. But it can cause problems, whene.g., in some biotech-nological processesmaterial systems are involved that exhibit non-Newtonianviscosity behavior. Then the shear stress exerted by the stirrer causes a viscosityfield.

Although most physical properties (e.g., viscosity, density, heat conductiv-ity and capacity, surface tension) must be regarded as variable, it is particularlythe value of viscosity that can be varied by many orders of magnitude under cer-tain process conditions [5,11]. In the following, dimensional analysis will be ap-plied, via examples, to describe the temperature dependency of the density undviscosity of non-Newtonian fluids as influenced by the shear stress.

A. Dimensionless Representation of the Material FunctionSimilar behavior of a certain physical property common to different material sys-tems can only be visualized by dimensionless representation of the material func-tion of that property (here the density ). It is furthermore desirable to formulate


this function as uniformly as possible. This can be achieved by the standard rep-resentation [6] of the material function in which a standardized transformation ofthe material function (T ) is defined in such a way that the expression produced,

/0 {0(T T0)} (28)meets the requirement

(0) (0) 1

where 0 10 T0 temperature coefficient of the density and 0 (T0).T0 is any reference temperature.

Figure 4A shows the dependency (T) for four different liquids, and Figure4B depicts the standard representation of this behavior. This confirms thatpropene, toluene, and CCl4 behave similarly with regard to (T), whereas waterbehaves differently. This implies that water cannot be used in model experimentsif one of the other three liquids will be employed in the industrial plant.

B. Pi Set for Temperature-Dependent Physical PropertiesThe type of dimensionless representation of the material function affects the (ex-tended) pi set within which the process relationship is formulated (for more infor-mation see Ref. 5). When the standard representation is used, the relevance listmust include the reference density 0 instead of and incorporate two additionalparameters 0, T0. This leads to two additional dimensionless numbers in the pro-cess characteristics. With regard to the heat transfer characteristics of a mixingvessel or a smooth straight pipe, Eq. (27), it now follows that

Nu (Re0, Pr0, 0T, T/T0) (29)where the subscript zero in Re and Pr denotes that these two dimensionless num-bers are to be formed with 0 (which is the numerical value of at T0).

If we consider that the standard transformation of the material function canbe expressed invariantly with regard to the reference temperature T0 (Fig. 4b),then the relevance list is extended by only one additional parameter, 0. This, inturn, leads to only one additional dimensionless number. For the foregoing prob-lem it now follows that

Nu (Re0, Pr0, 0T ) (30)

C. Non-Newtonian LiquidsThe main characteristics of Newtonian liquids is that simple shear flow (e.g., Cou-ette flow) generates shear stress that is proportional to the shear rate dv/dy

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Figure 4 (A) Temperature dependency of the density, (T), for four different liquids. (B)The standard representation of the behavior (T) for the same liquids.

[sec1]. The proportionality constant, the dynamic viscosity , is the only mate-rial constant in Newtons law of motion:

(31) depends only on temperature.

In the case of non-Newtonian liquids, depends on as well. These liquidscan be classified into various categories of materials depending on their flow be-havior: () flow curve and () viscosity curve.

D. Pseudoplastic FluidsAn extensive class of non-Newtonian fluids is formed by pseudoplastic fluidswhose flow curves obey the so-called power law:

Km eff K(m1) (32)These liquids are known as Ostwaldde Waele fluids. Figure 5 depicts a typicalcourse of such a flow curve. Figure 6 shows a dimensionless standardized mate-rial function of two pseudoplastic fluids often used in biotechnology. It proves thatthey behave similarly with respect to viscosity behavior under shear stress.

E. Viscoelastic LiquidsAlmost every biological solution of low viscosity [but also viscous biopolymerslike xanthane and dilute solutions of long-chain polymers, e.g., carboxymethyl-cellulose (CMC), polyacrylamide (PAA), and polyacrylnitrile (PAN)] displays

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Figure 5 Typical flow behavior of pseudoplastic fluids.

not only viscous but also viscoelastic flow behavior. These liquids are capable ofstoring a part of the deformation energy elastically and reversibly. They evade me-chanical stress by contracting like rubber bands. This behavior causes a secondaryflow that often runs contrary to the flow produced by mass forces (e.g., the liquidclimbs the shaft of a stirrer, the so-called Weissenberg effect).

Elastic behavior of liquids is characterized mainly by the ratio of first dif-ferences in normal stress, N1, to the shear stress, . This ratio, the Weissenbergnumber Wi N1/, is usually represented as a function of the rate of shear . Fig-ure 7 depicts flow curves of some viscoelastic fluids, and Figure 8 presents a di-mensionless standardized material function of these fluids. It again verifies thatthey behave similarly with respect to viscoelastic behavior under shear stress.

F. Pi Set for Non-Newtonian FluidsThe transition from a Newtonian to a non-Newtonian fluid results in the follow-ing consequences regarding the extension of the pi set.

1. All pi numbers of the Newtonian case also appear in the non-Newtoniancase, whereby is exchanged by a quantity H with the dimension ofviscosity (mostly 0).


Figure 6 Dimensionless standardized material function of some pseudoplastic fluidsused as model substances in biotechnological research. (From Ref. 2.)

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Figure 7 Flow curves of viscoelastic fluids often used in the biotechnological research(PAA, CMC). (From Ref. 12.)

Figure 8 Dimensionless standardized material function of the fluids in Figure 7, verify-ing the similar viscoelastic behavior under shear stress. (From Ref. 12.)

2. An additional pi number appears that contains a quantity with the di-mension of time (mostly 1/0).

3. The pure material numbers are extended by rheol.

The table below illustrates this using the example chosen at the beginningof this chapter, namely the heat transfer characteristics of a mixing vessel or asmooth straight pipe, Eq. (27). It shows the complete set of pi numbers for a tem-perature independent (a) and temperature dependent (b) viscosity of a Newtonianand a non-Newtonian fluid.

(33)Newtonian fluid non-Newtonian fluid

a Nu, Re, Pr Nu, ReH, PrH, v/L, rheolb Nu, Re0, Pr0, 0 T Nu, ReHo, PrHo, v0 /L, HoT, /H, rheol

In (b), the pi numbers w/ and HoT as well as /H, have to be added ( ln /T ). Besides this, completely other phenomena can occur (e.g., creeping ofa viscoelastic liquid on a rotating stirrer shaft opposite to gravitythe so-calledWeissenberg effect) that require additional parameters (in this case g) to be incor-porated into the relevance list.

VII. DETERMINATION OF OPTIMUM PROCESSCONDITIONS BY COMBINING PROCESSCHARACTERISTICS

The next example shows how a meaningful combination of appropriate processcharacteristics makes it possible to gain the information necessary for the opti-mization of the process in question.Example 5: Optimum Conditions for the Homogenization of Liquid Mixtures.The homogenization of miscible liquids is one of most frequent mixing opera-tions. It can be executed properly if the power characteristics and the mixing-timecharacteristics of the stirrer in question are known. If these characteristics areknown for a series of common stirrer types under favorable installation condi-tions, one can go on to consider optimum operating conditions by asking the fol-lowing question: Which type of stirrer operates within the requested mixing time with the lowest power consumption P and hence the minimum mixing work (P min) in a given material system and a given vessel (vessel diameter D)?Example 5.1: Power Characteristics of a Stirrer. The relevance list for this taskconsists of the target quantity (mixing power P) and the following parameters:stirrer diameter d, density and kinematic viscosity of the liquid, and stirrer


speed n:

{P; d; , ; n} (34)By choosing the dimensional matrix

d n P

Mass M 1 0 0 1 0Length L 3 1 0 2 2

Time T 0 0 1 3 1core matrix residual matrix

only one linear transformation is necessary to obtain the unity matrix:

d n P

M 1 0 0 1 03M L 0 1 0 5 2

T 0 0 1 3 1unity matrix residual matrix

The residual matrix consists of only two parameters, so only two pi numbersresult:

1 1n

P3d5

nP3d5 Ne (Newton number)

2 0n

1d2

n

d2 Re1 (Reynolds number)

The process characteristics

Ne (Re) (35)for three well-known, slowly rotating stirrers (leaf, frame, and cross-beam stirrers)is presented in Fig. 9.

1. In the range Re 20, the proportionality Ne Re1 is found, thus re-sulting in the expression NeRe P/(n2d3) const. Density is irrele-vant herewe are dealing with the laminar flow region.

2. In the range Re 50 (vessel with baffles) or Re 5 104 (unbaf-fled vessel), the Newton number Ne P/(n3d5) remains constant. Inthis case, viscosity is irrelevantwe are dealing with a turbulent flowregion.

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3. Understandably, the baffles do not influence the power characteristicswithin the laminar flow region, where viscosity forces prevent ro

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